I gave the first section a try:

My darling,

You have lost me from now on

I ripped your love from my heart 

For whoever you want, open a place in your heart

Let happiness stay in your shadow.

If someone gets the chance, hope to hear if these are OK?

1. We are not going to Bogota this time.

Bu sefer Bogota´ya gitmeyeceğiz.

2. We saw cactus on the way to Pasto.

 Pasto´ya giderken kaktüsü gördük. (Kaktüs is not a definite object in this sentence. If it was definite, your sentence would be correct. "The" would make it definite. But there is no "the" in your sentence.You should study definite and indefinite objects.

http://www.turkishlanguage.co.uk/defarts.htm

If you want to make kaktüs plural instead of adding ler/lar, use birçok. Pasto yolunda birçok kaktüs gördük.

3. Did you hear the girl singing in the street last night?

Dün gece  sokakta şarkı söyleyen kızı duydunuz mi?

4. I am staying here for one more week.

Burada bir hafta daha kaliyorum.

This is all very confusing to me, because I think I really know this construction only in the form of "-eyim" and "-elim." Or "let me" and "let us." If I´m understanding you correctly, this is what my old Turkish grammar book calls the "present optative-subjective verb form." I have to admit I didn´t pay close attention to this lesson because it said that the first person singular and first person plural (which I already knew from a Turkish Class lesson on the imperative) were used, but that other forms were not so much. (For instance, in third person, "anlasın" -- or the third person imperative -- would be used more, yes?) So, I was very familiar with "-eyim" and "-elim," but I didn´t worry about "-esin." But that´s where that extra "a" and the buffer y come from, yes? This was what I was not recognizing. So the construction is "anlaya-" (which is the gerundive, not the verb root) plus "-sın." And this "present optative-subjective verb form" expresses "may" or "might." So my puzzling sentence becomes something like "Who understood it that you might (or should) understand it?" In a more literal form, I mean. Your translations make much more sense, gokuyum.

Türkçe çok zor! See how hard I have to think about all this! And I still may be confused.

Yardımın için teşekkür ederim. 

"Kim anlamış ki sen anlayasın böyle?"

Merhaba! I found this line in a song, and from context, I think it means something like: "Who thought that your understanding would be like this?" But please correct me, and please explain what is going on here grammatically. 

Is "anlamış" the indefinite past? And what is "anlayasın"? I cannot figure out this ending. It´s not a possessive, is it? I see "sen" but not "senin." I´m stuck. İmdat!

Çok teşekkür ederim! 

şiş kebap. et or tavuk, doesnt matter, both delicious.